%%%%(c) COPYRIGHT NOTICE%FOLDUP
%%%%(c)
%%%%(c)  This file is a portion of the source for the textbook
%%%%(c)
%%%%(c)    Numerical Methods Course Notes,
%%%%(c)    Copyright 2004-2010 by Steven E. Pav
%%%%(c)
%%%%(c)  See the file COPYING.txt for copying conditions
%%%%(c)
%%%%(c)%UNFOLD

%cvs%stuff%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%FOLDUP
% $Id: sepmesh.sty 3 2005-11-08 20:56:42Z spav $
% $Log: sepmesh.sty,v $
% Revision 1.10  2005/04/19 23:00:10  spav
% some changes
%
% Revision 1.9  2005/03/17 18:11:39  spav
% minor cosmetic changes
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%UNFOLD

%%throat clearing%FOLDUP
\typeout{-- sepmesh.sty}
\typeout{-- N© 2004-2010 Steven E. Pav}
%UNFOLD

% series for graph edges
\def\@MSHedgeseries#1{\ifcase#1\or e \or f \or g\else\@ctrerr\fi\relax}
\providecommand{\edUL}[3]{\ensuremath{\neUL{\@MSHedgeseries{#1}}{#2}{#3}}\xspace}

% series for graph points
\def\@MSHpointseries#1{\ifcase#1\or p \or q \or t\else\@ctrerr\fi\relax}
\providecommand{\ptUL}[3]{\ensuremath{\neUL{\@MSHpointseries{#1}}{#2}{#3}}\xspace}

% respectively, the DT, line segment, local feature size, simplicial set,
% three kinds of polytopes, simplex and bound for skinnyes.
\providecommand{\DLNY}{{D}elaunay\xspace}
\providecommand{\DR}{\DLNY Refinement\xspace}
\providecommand{\DT}{\DLNY Triangulation\xspace}
\providecommand{\DTs}{\DLNY Triangulations\xspace}
\providecommand{\CDT}{Constrained \DT}
\providecommand{\CDTs}{Constrained \DTs}
\providecommand{\CoDT}{Conforming \DT}
\providecommand{\CoDTs}{Conforming \DTs}
\providecommand{\DTet}{\DLNY Tetrahedralization\xspace}
\providecommand{\DTets}{\DLNY Tetrahedralizations\xspace}
\providecommand{\CoDTet}{Conforming \DTet}
\providecommand{\CoDTets}{Conforming \DTets}
%\providecommand{\VRNOI}{{V}orono\"{\i}\xspace}
\providecommand{\VRNOI}{{V}oronoi\xspace}
\providecommand{\VD}{\VRNOI Diagram\xspace}

\providecommand{\lseg}[2]{\ensuremath{\wrapParens{#1, #2}}}
\providecommand{\cseg}[2]{\ensuremath{\widetilde{#1, #2}}}
%\providecommand{\lseg}[2]{\ensuremath{\overline{#1 #2}}\xspace}
\providecommand{\CD}[1]{\ensuremath{\left(#1\right)}\xspace}

\providecommand{\sset}[1][l]{\ensuremath{\mathcal{T}_{#1}}\xspace}
\providecommand{\ptope}[1][l]{\ensuremath{\mathcal{P}_{#1}}\xspace}
\providecommand{\qtope}[1][l]{\ensuremath{\mathcal{Q}_{#1}}\xspace}
\providecommand{\rtope}[1][l]{\ensuremath{\mathcal{R}_{#1}}\xspace}
\providecommand{\xtope}[1][l]{\ensuremath{\mathcal{P}_{#1}^*}\xspace}
\providecommand{\ntope}[2][l]{\ensuremath{\mathcal{P}_{#1}^{#2}}\xspace}
\providecommand{\smplx}[1][l]{\ensuremath{S_{#1}}\xspace}
\providecommand{\splx}{\smplx[]}
\providecommand{\tmplx}[1][l]{\ensuremath{T_{#1}}\xspace}
\providecommand{\xmplx}[1][l]{\ensuremath{S_{#1}^*}\xspace}
\providecommand{\nmplx}[2][l]{\ensuremath{S_{#1}^{#2}}\xspace}
\providecommand{\cmplx}[1][]{\ensuremath{\mathscr{S}_{#1}}\xspace}
\providecommand{\pgon}[1][]{\ensuremath{\mathscr{P}_{#1}}\xspace}
\providecommand{\skbd}[1][l]{\ensuremath{\rho_{#1}}\xspace}
\providecommand{\link}[2][\cmplx]{\ensuremath{\operatorname{link}_{#1}\left(#2\right)}\xspace}

%Bottom,Top,LocalFeature size: BTLF
\providecommand{\BTLF}[3]{\ensuremath{\mathrm{lfs}_{#1}^{#2}\wrapNeParens{#3}}\xspace}
\providecommand{\ulf}[2][{}]{\BTLF{}{#1}{#2}}
\providecommand{\Lf}[2][{}]{\BTLF{*}{#1}{#2}}
\providecommand{\lf}[2][{}]{\BTLF{#1}{}{#2}}
\providecommand{\lfs}[1][l]{\BTLF{#1}{}{}}
\providecommand{\nlf}[2][{}]{\BTLF{#1}{\prime}{#2}}
\providecommand{\nnlf}[2][{}]{\BTLF{#1}{\prime\prime}{#2}}

%Bottom,Top,NearestFeatureDistance: BTNF
\providecommand{\BTNF}[3]{\ensuremath{\mathrm{nfd}_{#1}^{#2}\wrapNeParens{#3}}\xspace}
\providecommand{\unf}[2][{}]{\BTNF{}{#1}{#2}}
\providecommand{\Nf}[2][{}]{\BTNF{*}{#1}{#2}}
\providecommand{\nf}[2][{}]{\BTNF{#1}{}{#2}}
\providecommand{\nfd}[1][l]{\BTNF{#1}{}{}}
\providecommand{\nnf}[2][{}]{\BTNF{#1}{\prime}{#2}}
\providecommand{\nnnf}[2][{}]{\BTNF{#1}{\prime\prime}{#2}}

%Bottom,Top,Nearest Neighbor: BTNN
\providecommand{\BTNN}[3]{\ensuremath{\mathrm{nn}_{#1}^{#2}\!\wrapNeParens{#3}}\xspace}
\providecommand{\nn}[2][{}]{\BTNN{#1}{}{#2}}

%Bottom,Top,distance function: BTDF
\providecommand{\BTDF}[3]{\ensuremath{\mathrm{d}_{#1}^{#2}\wrapNeParens{#3}}\xspace}

% rays, power of sphere and point, circumboll, border, cylinder
%\providecommand{\psD}[2]{\ensuremath{\operatorname{d}\left(#1,#2\right)}}
\providecommand{\ray}[1]{\ensuremath{\stackrel{\rightarrow}{#1}}\xspace}
\providecommand{\segray}[2]{\ray{\lseg{1}{2}}}
\providecommand{\hnorm}[1]{\ensuremath{\eta(#1)}\xspace}

\providecommand{\psD}[2]{\ensuremath{\mathscr{P}\left(#1,#2\right)}\xspace}
\providecommand{\bDom}[2]{\ensuremath{\left\{#1 \sqsubset #2\right\}}\xspace}
\providecommand{\bDomeq}[2]{\ensuremath{\left\{#1 \sqsubseteq #2\right\}}\xspace}
\providecommand{\cDom}[2]{\bDom{\cball{#1}}{\cball{#2}}}
\providecommand{\cDomeq}[2]{\bDomeq{\cball{#1}}{\cball{#2}}}

\providecommand{\beq}[2]{\ensuremath{\left\{#1 = #2\right\}}\xspace}
\providecommand{\radax}[2]{\beq{#1}{#2}}
\providecommand{\cball}[1]{\ensuremath{\mathscr{C}\wrapNeParens{#1}}\xspace}
\providecommand{\smpball}[1][l]{\cball{\smplx[#1]}}
\providecommand{\tmpball}[1][l]{\cball{\tmplx[#1]}}
\providecommand{\xmpball}[1][l]{\cball{\xmplx[#1]}}
\providecommand{\nmpball}[2][l]{\cball{\nmplx[#1]{#2}}}

\providecommand{\cyl}[1]{\ensuremath{\operatorname{Cyl}(#1)}\xspace}
\providecommand{\cylG}[2]{\ensuremath{\operatorname{Cyl}_{#2}(#1)}\xspace}
\providecommand{\cylp}[1]{\ensuremath{\operatorname{Cyl}^{+}(#1)}\xspace}
\providecommand{\cyln}[1]{\ensuremath{\operatorname{Cyl}^{-}(#1)}\xspace}
\providecommand{\cylGp}[2]{\ensuremath{\operatorname{Cyl}_{#2}^{+}(#1)}\xspace}
\providecommand{\cylGn}[2]{\ensuremath{\operatorname{Cyl}_{#2}^{-}(#1)}\xspace}
\providecommand{\cylpp}[1]{\ensuremath{\operatorname{Cyl}^{++}(#1)}\xspace}
\providecommand{\cylnn}[1]{\ensuremath{\operatorname{Cyl}^{--}(#1)}\xspace}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% laguerre/voronoi dominating region
\providecommand{\LVdom}[2][\Balset]{\ensuremath{\operatorname{Dom}_{#1}(#2)}\xspace}
\providecommand{\LVR}[2][\Balset]{\ensuremath{\operatorname{LR}_{#1}(#2)}\xspace}
\providecommand{\chull}[1]{\ensuremath{\mathcal{CH}\left(#1\right)}\xspace}
\providecommand{\bptope}[1][l-j]{\rtope[#1]}
\providecommand{\bptopes}[1][j]{\ensuremath{\Delta_{#1}}\xspace}
\providecommand{\bsmplx}[1][l-j]{\tmplx[#1]}
\providecommand{\bsmplxs}[1][j]{\ensuremath{\Sigma_{#1}}\xspace}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% representatives for various things
\providecommand{\HSSrep}{\ensuremath{\mathcal{H}}\xspace}
\providecommand{\Ssetsrep}{\ensuremath{\mathbf{\bar{T}}}\xspace}
\providecommand{\ConfQrep}{\ensuremath{\mathcal{Q}}\xspace}
\providecommand{\Scmprep}{\ensuremath{\mathscr{S}}\xspace}
%\providecommand{\Qsprep}{\ensuremath{\mathcal{Q}}\xspace}
%\providecommand{\Ssprep}{\ensuremath{\mathcal{S}}\xspace}
%\providecommand{\Ruprep}{\ensuremath{\mathcal{R}}\xspace}
%\providecommand{\VtxTabrep}{\ensuremath{\mathcal{V}}\xspace}
\providecommand{\SknyTrep}{\ensuremath{\mathcal{X}}\xspace}
\providecommand{\BassTrep}{\ensuremath{\mathcal{B}}\xspace}
\providecommand{\PLSrep}{\ensuremath{\mathscr{W}}\xspace}
\providecommand{\PLCrep}{\ensuremath{\mathscr{W}}\xspace}
\providecommand{\ptoperep}{\ensuremath{\mathcal{P}}}
\providecommand{\trirep}{\ensuremath{\mathcal{T}}}
\providecommand{\skelrep}{\ensuremath{\mathscr{S}}\xspace}
%polytope, edge, segs, triangles, faces, arcs sets
\providecommand{\ptpsetrep}{\ensuremath{\mathscr{B}}\xspace}
\providecommand{\edgsetrep}{\ensuremath{\mathscr{E}}\xspace}
\providecommand{\segsetrep}{\ensuremath{\mathscr{S}}\xspace}
\providecommand{\trisetrep}{\ensuremath{\mathscr{T}}\xspace}
\providecommand{\facsetrep}{\ensuremath{\mathscr{F}}\xspace}
\providecommand{\arcsetrep}{\ensuremath{\mathscr{A}}\xspace}
%vertex, point, curve sets.
\providecommand{\vtxsetrep}{\ensuremath{\mathscr{V}}\xspace}
\providecommand{\pntsetrep}{\ensuremath{\mathscr{P}}\xspace}
\providecommand{\crvsetrep}{\ensuremath{\mathscr{C}}\xspace}
%circles?, trns?
\providecommand{\crcsetrep}{\ensuremath{\mathscr{C}}\xspace}
\providecommand{\trnsetrep}{\ensuremath{\mathscr{T}}\xspace}

\providecommand{\pnts}{\pntsetrep}
\providecommand{\segs}{\segsetrep}
\providecommand{\tris}{\trisetrep}
\providecommand{\crvs}{\crvsetrep}
\providecommand{\trng}{\trnsetrep}
\providecommand{\ptps}{\ptpsetrep}
\providecommand{\vtxs}{\vtxsetrep}
\providecommand{\edgs}{\edgsetrep}
\providecommand{\facs}{\facsetrep}
\providecommand{\crcs}{\crcsetrep}
\providecommand{\arcs}{\arcsetrep}
\providecommand{\pls}{\PLSrep}
\providecommand{\plc}{\PLCrep}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%input degree, point neighbor, fans, fanlengths, fanrings.
\providecommand{\idg}[2][{}]{\ensuremath{\operatorname{dg}_{#1}\left(#2\right)}\xspace}
\providecommand{\pnbr}[2][{}]{\ensuremath{\mathcal{N}_{#1}\left(#2\right)}\xspace}
%\providecommand{\fan}[2][{}]{\ensuremath{\mathscr{F}_{#1}\left(#2\right)}\xspace}
%\providecommand{\fln}[1]{\ensuremath{\operatorname{Fl}\left(#1\right)}\xspace}
%\providecommand{\frng}[2][0]{\ensuremath{\mathcal{FR}_{#1}\left(#2\right)}\xspace}
\providecommand{\fan}[3][{}]{\ensuremath{\mathscr{F}_{#1}^{#2}\left(#3\right)}\xspace}
\providecommand{\fln}[2]{\ensuremath{\operatorname{Fl}^{#1}\left(#2\right)}\xspace}
\providecommand{\frng}[3][0]{\ensuremath{\mathcal{FR}_{#1}^{#2}\left(#3\right)}\xspace}

\providecommand{\Bals}{\ensuremath{\mathcal{B}}\xspace}
\providecommand{\Balset}{\ensuremath{\Omega}\xspace}

\providecommand{\HSSvtxs}{\ensuremath{\mathcal{V}_{\HSSrep}}\xspace}
\providecommand{\smpvtxs}{\ensuremath{\mathcal{V}_s}\xspace}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%typical vertex set, and minimum distance?
\providecommand{\vtxSet}[1]{\ensuremath{\mathbf{V}\left(#1\right)}\xspace}
\providecommand{\minToVtx}[2]{\ensuremath{\mathfrak{m}\left(#1, #2\right)}\xspace}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%the delaunay operator, and cavity junk
\providecommand{\dt}[1][{}]{\ensuremath{\mathcal{D}\wrapNeParens{#1}}\xspace}
\providecommand{\vd}[1][{}]{\ensuremath{\mathcal{V}\wrapNeParens{#1}}\xspace}
\providecommand{\cavt}{\ensuremath{\mathcal{C}}\xspace}
\providecommand{\cavtb}{\ensuremath{\mathcal{C}_b}}
\providecommand{\cavtS}{\ensuremath{\mathcal{C}_S}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% parent relation
\providecommand{\prnt}[2]{\ensuremath{\left(#1\right) \stackrel{b}{\leadsto} \left(#2\right)}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% angle stuff

\providecommand{\Minin}[1][{}]{\ensuremath{\anUL{1}{*}{#1}}\xspace}
\providecommand{\minin}[2]{\Minin[#1\lMr{#2}]}

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